MATH 243, LECTURE 9
Probability is nothing but common sense reduced to calculation.
Pierre Simon Laplace
1. Basic probability
The theory of probability let’s us answer questions such as “what are the chances that at random these
ten people will get well (in our study) but those ten people will not?” In general probability theory helps
in predict what will happen for a random process over many trials.
“Randomness” is a more difficult concept than you would think, but there are some clear examples.
Observationally, the outcome of a coin toss is random since we can’t predict heads or tails.
Simultaneously, however, there is a long-term pattern to the outcome. If enough tosses are made,
approximately half will be heads, and half tails.
This does not mean that every “heads” will be followed by a “tail.” Just that if you toss long enough,
you expect to see half heads and half tails.
Famous question: If you toss a coin 5 times in a row and get heads every time, what is the probability
of getting tails the next time?
Same kind of question: If you have 4 girls already, are you more likely to have a boy as your next child?
Notation 1 (Notation by example).•We let Xbe the result of a coin toss. It can be Hor T.
{H, T }is called the sample space.
•We write P(X=H) = .5to mean the probability of the result of the coin toss being heads is .5.
•P(X=H or X =T) = 1 since those are the only possibilities.
Example 2. Let Xbe the result of two successive coin tosses. What is the sample space? What are the
probabilities of each member of the sample space?
Notice that if we add the probability for each event in our sample space we get 1.
Example 3. Basically same example, different sample space: answer the same questions when Xis the
number of heads in two successive coin tosses.
Probability is difficult to formulate in precise mathematical terms (though it is well worth doing so!),
but we can make do with some intuitive characterizations.
Definition 4. Let Xbe the outcome of some event.
•The outcome of Xis a random event if we cannot predict it, but the distribution of such outcomes
has a regular pattern over many repetitions.
•Xis called a random variable.
•The sample space is the collection all possible values Xcould take.
•The probability of a certain outcome is the proportion of times it occurs after many repetitions.
Example 5. Back to the coin example, does a probability of 1
2for heads mean that if we flip the coin
6 times, we get 3 heads? Does this even mean that over Nflips, as Ngets large the number of heads
approaches N/2?
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